Go Nittany Lions!!
Ex 1) Circular motion horizontal surface w/ friction
A car with mass M moves in a circle of radius R on a asphalt road (µ) with a speed V. Find V
F_{c} = mv^{2}/r = µmg
v^{2}/r = µg
v = (µgr)^{1}
Ex 2) Mass on a String  Vertical Circle
Mass M attached to a string moves at constant speed of v in a vertical circle of radius r. Find string tension T_{top} and T_{bottom}
Top Free Body Diagram
Make center of circle + F_{net} = ma F_{net }= Mg + T_{top}
Mv^{2}/r = Mg + T_{top} M[v^{2}/r  g] = T_{top}
F_{net} = ma F_{net }= T_{bottom }_{ }Mg
Mv^{2}/r = T_{bottom }_{ }Mg M[v^{2}/r +_{ }g] = T_{bottom }
Mv^{2}/r = Mg + T_{top} Mv^{2}/r = T_{bottom }_{ }Mg
Mg + T_{top }= T_{bottom}  Mg
2Mg + T_{top }= T_{bottom}
Ex 3) Object mass M moves at constant speed V at the end of string in a circle r. What is the minimum speed needed to maintain the object's circular motion?
What is T_{top} at minimum speed?
T_{top } = 0
Free Body Diagram
Make center of circle + F_{net} = ma F_{net }= Mg + T_{top}
Mv^{2}/r = Mg
When T_{top }= 0 v^{2}/r = g
Ex 4) Rollercoaster  velocity changes with height
Car of mass M, circle radius r, Find Normal Force at top
Freebody diagram at top: F_{net} = mv^{2}/r
F_{n} + mg = mv^{2}/r
F_{n} = [mv^{2}/r]  mg
What minimum velocity is required for the roller coaster to prevent the passengers from falling off?
Minimum v occurs when F_{n} = 0 F_{n} = [mv^{2}/r]  mg
[mv^{2}/r] = mg
v^{2}/r = g
İTony Mangiacapre.,
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